If \( P=\begin{bmatrix}0 & 1\\1 & 1\end{bmatrix} \) and \( Q=\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix} \), then which of the following are correct?
(A) \( P+Q \) is a singular matrix.
(B) \( P-Q \) is a symmetric matrix.
(C) \( P \) and \( Q \) are non-singular matrices.
(D) \( (P+Q)(P-Q)\ne P^2-Q^2 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
Adding and subtracting the matrices:
\( \displaystyle P+Q=\begin{bmatrix}0 & 0\\2 & 1\end{bmatrix} \)
\( \displaystyle P-Q=\begin{bmatrix}0 & 2\\0 & 1\end{bmatrix} \)
The determinants are
\( |P+Q|=0-0=0 \)
\( |P|=0-1=-1 \)
\( |Q|=0+1=1 \)
Expanding, \( (P+Q)(P-Q)=P^2-PQ+QP-Q^2 \), which equals \( P^2-Q^2 \) only if PQ = QP. Here
\( \displaystyle PQ=\begin{bmatrix}1 & 0\\1 & -1\end{bmatrix} \)
\( \displaystyle QP=\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix} \)
So |P + Q| = 0, P − Q is not symmetric, |P| = −1, |Q| = 1, and PQ ≠ QP.
Checking the statements
(A) \( P+Q \) is a singular matrix. — Correct. \( |P+Q|=0 \).
(B) \( P-Q \) is a symmetric matrix. — Incorrect. Its (1, 2) entry is 2 but its (2, 1) entry is 0.
(C) \( P \) and \( Q \) are non-singular matrices. — Correct. \( |P|=-1 \) and \( |Q|=1 \), both non-zero.
(D) \( (P+Q)(P-Q)\ne P^2-Q^2 \) — Correct. \( PQ\ne QP \), so the product is not \( P^2-Q^2 \).